Indexed Family

In mathematics, an indexed family is a collection of values that are associated with indexes. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers.

Formally, an indexed family is the same thing as a function. A function with domain J and codomain X is equivalent to a family of elements of X indexed by elements of J. The only difference is that indexed families are thought of as collections instead of as functions. A value is considered to be an element of a family whenever it is an element of the image of the family's underlying function.

When a function f : JX is treated as a family, J is called the index set of the family, the functional image f(j) for jJ is denoted xj, and the mapping f is denoted {xj}jJ or simply {xj}.

Next, if the set X is the power set of a set U, then the family {xj}jJ is called a family of sets indexed by J .

Read more about Indexed FamilyMathematical Statement, Functions, Sets and Families, Examples, Operations On Families, Subfamily, Usage in Category Theory

Other articles related to "indexed family, indexed":

Indexed Family - Usage in Category Theory
... A diagram is a functor giving rise to an indexed family of objects in a category C, indexed by another category J, and related by morphisms depending on two indices ...
Egorov's Theorem - Generalizations - Korovkin's Version - Proof
... Consider the indexed family of sets whose index set is the set of natural numbers m, defined as follows Obiviously and therefore there is a natural number m0 such that putting A0,m0=A0 the following relation ...

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